![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > dvcosre | Structured version Visualization version GIF version |
Description: The real derivative of the cosine. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
Ref | Expression |
---|---|
dvcosre | ⊢ (ℝ D (𝑥 ∈ ℝ ↦ (cos‘𝑥))) = (𝑥 ∈ ℝ ↦ -(sin‘𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reelprrecn 10618 | . . 3 ⊢ ℝ ∈ {ℝ, ℂ} | |
2 | cosf 15470 | . . 3 ⊢ cos:ℂ⟶ℂ | |
3 | ssid 3937 | . . 3 ⊢ ℂ ⊆ ℂ | |
4 | nfcv 2955 | . . . . . 6 ⊢ Ⅎ𝑥ℝ | |
5 | nfrab1 3337 | . . . . . 6 ⊢ Ⅎ𝑥{𝑥 ∈ ℂ ∣ -(sin‘𝑥) ∈ V} | |
6 | 4, 5 | dfss2f 3905 | . . . . 5 ⊢ (ℝ ⊆ {𝑥 ∈ ℂ ∣ -(sin‘𝑥) ∈ V} ↔ ∀𝑥(𝑥 ∈ ℝ → 𝑥 ∈ {𝑥 ∈ ℂ ∣ -(sin‘𝑥) ∈ V})) |
7 | recn 10616 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ) | |
8 | 7 | sincld 15475 | . . . . . . . 8 ⊢ (𝑥 ∈ ℝ → (sin‘𝑥) ∈ ℂ) |
9 | 8 | negcld 10973 | . . . . . . 7 ⊢ (𝑥 ∈ ℝ → -(sin‘𝑥) ∈ ℂ) |
10 | elex 3459 | . . . . . . 7 ⊢ (-(sin‘𝑥) ∈ ℂ → -(sin‘𝑥) ∈ V) | |
11 | 9, 10 | syl 17 | . . . . . 6 ⊢ (𝑥 ∈ ℝ → -(sin‘𝑥) ∈ V) |
12 | rabid 3331 | . . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∈ ℂ ∣ -(sin‘𝑥) ∈ V} ↔ (𝑥 ∈ ℂ ∧ -(sin‘𝑥) ∈ V)) | |
13 | 7, 11, 12 | sylanbrc 586 | . . . . 5 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ {𝑥 ∈ ℂ ∣ -(sin‘𝑥) ∈ V}) |
14 | 6, 13 | mpgbir 1801 | . . . 4 ⊢ ℝ ⊆ {𝑥 ∈ ℂ ∣ -(sin‘𝑥) ∈ V} |
15 | dvcos 24586 | . . . . 5 ⊢ (ℂ D cos) = (𝑥 ∈ ℂ ↦ -(sin‘𝑥)) | |
16 | 15 | dmmpt 6061 | . . . 4 ⊢ dom (ℂ D cos) = {𝑥 ∈ ℂ ∣ -(sin‘𝑥) ∈ V} |
17 | 14, 16 | sseqtrri 3952 | . . 3 ⊢ ℝ ⊆ dom (ℂ D cos) |
18 | dvres3 24516 | . . 3 ⊢ (((ℝ ∈ {ℝ, ℂ} ∧ cos:ℂ⟶ℂ) ∧ (ℂ ⊆ ℂ ∧ ℝ ⊆ dom (ℂ D cos))) → (ℝ D (cos ↾ ℝ)) = ((ℂ D cos) ↾ ℝ)) | |
19 | 1, 2, 3, 17, 18 | mp4an 692 | . 2 ⊢ (ℝ D (cos ↾ ℝ)) = ((ℂ D cos) ↾ ℝ) |
20 | ffn 6487 | . . . . . . 7 ⊢ (cos:ℂ⟶ℂ → cos Fn ℂ) | |
21 | 2, 20 | ax-mp 5 | . . . . . 6 ⊢ cos Fn ℂ |
22 | dffn5 6699 | . . . . . 6 ⊢ (cos Fn ℂ ↔ cos = (𝑥 ∈ ℂ ↦ (cos‘𝑥))) | |
23 | 21, 22 | mpbi 233 | . . . . 5 ⊢ cos = (𝑥 ∈ ℂ ↦ (cos‘𝑥)) |
24 | 23 | reseq1i 5814 | . . . 4 ⊢ (cos ↾ ℝ) = ((𝑥 ∈ ℂ ↦ (cos‘𝑥)) ↾ ℝ) |
25 | ax-resscn 10583 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
26 | resmpt 5872 | . . . . 5 ⊢ (ℝ ⊆ ℂ → ((𝑥 ∈ ℂ ↦ (cos‘𝑥)) ↾ ℝ) = (𝑥 ∈ ℝ ↦ (cos‘𝑥))) | |
27 | 25, 26 | ax-mp 5 | . . . 4 ⊢ ((𝑥 ∈ ℂ ↦ (cos‘𝑥)) ↾ ℝ) = (𝑥 ∈ ℝ ↦ (cos‘𝑥)) |
28 | 24, 27 | eqtri 2821 | . . 3 ⊢ (cos ↾ ℝ) = (𝑥 ∈ ℝ ↦ (cos‘𝑥)) |
29 | 28 | oveq2i 7146 | . 2 ⊢ (ℝ D (cos ↾ ℝ)) = (ℝ D (𝑥 ∈ ℝ ↦ (cos‘𝑥))) |
30 | 15 | reseq1i 5814 | . . 3 ⊢ ((ℂ D cos) ↾ ℝ) = ((𝑥 ∈ ℂ ↦ -(sin‘𝑥)) ↾ ℝ) |
31 | resmpt 5872 | . . . 4 ⊢ (ℝ ⊆ ℂ → ((𝑥 ∈ ℂ ↦ -(sin‘𝑥)) ↾ ℝ) = (𝑥 ∈ ℝ ↦ -(sin‘𝑥))) | |
32 | 25, 31 | ax-mp 5 | . . 3 ⊢ ((𝑥 ∈ ℂ ↦ -(sin‘𝑥)) ↾ ℝ) = (𝑥 ∈ ℝ ↦ -(sin‘𝑥)) |
33 | 30, 32 | eqtri 2821 | . 2 ⊢ ((ℂ D cos) ↾ ℝ) = (𝑥 ∈ ℝ ↦ -(sin‘𝑥)) |
34 | 19, 29, 33 | 3eqtr3i 2829 | 1 ⊢ (ℝ D (𝑥 ∈ ℝ ↦ (cos‘𝑥))) = (𝑥 ∈ ℝ ↦ -(sin‘𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 {crab 3110 Vcvv 3441 ⊆ wss 3881 {cpr 4527 ↦ cmpt 5110 dom cdm 5519 ↾ cres 5521 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 -cneg 10860 sincsin 15409 cosccos 15410 D cdv 24466 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ico 12732 df-icc 12733 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-fac 13630 df-bc 13659 df-hash 13687 df-shft 14418 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-sum 15035 df-ef 15413 df-sin 15415 df-cos 15416 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-lp 21741 df-perf 21742 df-cn 21832 df-cnp 21833 df-haus 21920 df-tx 22167 df-hmeo 22360 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-xms 22927 df-ms 22928 df-tms 22929 df-cncf 23483 df-limc 24469 df-dv 24470 |
This theorem is referenced by: itgsin0pilem1 42592 itgsinexplem1 42596 fourierdlem39 42788 |
Copyright terms: Public domain | W3C validator |